Dummit and Foote Exercise 1 on page 545 reads as follows:
Determine the splitting field and its degree over for .
------------------------------------------------------------------------------------------------------
I have started on the solution to this exercise as follows:
The two roots of are and
Thus the splitting field is
We note that since the product of and i must be in
The element is algebraic over and so is the minimal polynomial for over .
Thus the degree of over is the degree of = 4
Hence = 4
The element is also algebraic over and so is the minimal polynomial for over .
Thus the degree of over is the degree of = 4
Hence = 4
But where to from here - need to find the degree of the splitting field.
Can someone please confirm that my reasoning above is valid and show me the way forward from here?
Peter